NDA & NA Exam (II)- 2015 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2015 - Mathematics

NDA & NA Exam (II)- 2015 - Mathematics Defence Questions

11.
If \( A = \{x \in \mathbb{R} : x^{2} + 6x - 7 < 0\} \) and \( B = \{x \in \mathbb{R} : x^{2} + 9x + 14 > 0\} \), then which of the following is/are correct?
1. \( A \cap B = (-2, 1) \)
2. \( A \setminus B = (-7, -2) \)
Select the correct answer using the code given below.
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
A. 1 only
12.
A, B, C and D are four sets such that \( A \cap B = C \cap D = \phi \). Consider the following:
1. \( A \cup C \) and \( B \cup D \) are always disjoint.
2. \( A \cap C \) and \( B \cap D \) are always disjoint.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
B. 2 only
13.
If A is an invertible matrix of order n and k is any positive real number, then the value of \( [\det(kA)]^{-1} \det A \) is
A.
\( k^{-n} \)
B.
\( k^{-1} \)
C.
\( k^{n} \)
D.
nk
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ANSWER :
A. \( k^{-n} \)
14.
The value of the infinite product \( 6^{\frac{1}{2}} \times 6^{\frac{1}{2}} \times 6^{\frac{3}{8}} \times 6^{\frac{1}{4}} \times \ldots \) is
A.
6
B.
36
C.
216
D.
\( \infty \)
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ANSWER :
B. 36
15.
If the roots of the equation \( x^{2} - nx + m = 0 \) differ by 1, then
A.
\( n^{2} - 4m - 1 = 0 \)
B.
\( n^{2} + 4m - 1 = 0 \)
C.
\( m^{2} + 4n + 1 = 0 \)
D.
\( m^{2} - 4n - 1 = 0 \)
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ANSWER :
A. \( n^{2} - 4m - 1 = 0 \)
16.
If different words are formed with all the letters of the word ‘AGAIN’ and are arranged alphabetically among themselves as in a dictionary, the word at the 50th place will be
A.
NAAGI
B.
NAAIG
C.
IAAGN
D.
IAANG
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ANSWER :
B. NAAIG
17.
The number of ways in which a cricket team of 11 players be chosen out of a batch of 15 players so that the captain of the team is always included, is
A.
165
B.
364
C.
1001
D.
1365
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ANSWER :
C. 1001
18.
In the expansion of \( \left(\sqrt{x} + \frac{1}{3x^{2}}\right)^{10} \), the value of constant term (independent of x) is
A.
5
B.
8
C.
45
D.
90
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ANSWER :
A. 5
19.
The value of \( \sin^{2} 5^{\circ} + \sin^{2} 10^{\circ} + \sin^{2} 15^{\circ} + \sin^{2} 20^{\circ} + \ldots + \sin^{2} 90^{\circ} \) is
A.
7
B.
8
C.
9
D.
\( \frac{19}{2} \)
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ANSWER :
D. \( \frac{19}{2} \)
20.
On simplifying \( \frac{\sin^{3} A + \sin 3A}{\sin A} + \frac{\cos^{3} A - \cos 3A}{\cos A} \), we get
A.
\( \sin 3A \)
B.
\( \cos 3A \)
C.
\( \sin A + \cos A \)
D.
3
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ANSWER :
D. 3
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